Statistical Analysis of Clinical Data on a Pocket Calculator
Ton J. Cleophas • Aeilko H. Zwinderman
Statistical Analysis of Clinical Data on a Pocket Calculator Statistics on a Pocket Calculator
Prof. Ton J. Cleophas Department of Medicine Albert Schweitzer Hospital Dordrecht, The Netherlands and European College of Pharmaceutical Medicine, Lyon, France
[email protected] Prof. Aeilko H. Zwinderman Department of Epidemiology and Biostatistics Academic Medical Center Amsterdam, The Netherlands and European College of Pharmaceutical Medicine, Lyon, France
[email protected] ISBN 978-94-007-1210-2 e-ISBN 978-94-007-1211-9 DOI 10.1007/978-94-007-1211-9 Springer Dordrecht Heidelberg London New York © Springer Science+Business Media B.V. 2011 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
The time that statistical analyses, including analysis of variance and regression analyses, were analyzed by statistical laboratory workers, has gone for good, thanks to the availability of user-friendly statistical software. The teaching department, the educations committee, and the scientific committee of the Albert Schweitzer Hospital, Dordrecht, Netherlands, are pleased to announce that since November 2009 the entire staff and personal is able to perform statistical analyses with help of SPSS Statistical Software in their offices through the institution’s intranet. It is our experience as masters’ and doctorate class teachers of the European College of Pharmaceutical Medicine (EC Socrates Project) that students are eager to master adequate command of statistical software for carrying out their own statistical analyses. However, students often lack adequate knowledge of basic principles, and this carries the risk of fallacies. Computers cannot think, and can only execute commands as given. As an example, regression analysis usually applies independent and dependent variables, often interprets as causal factors and outcome factors. E.g., gender and age may determine the type of operation or the type of surgeon. The type of surgeon does not determine the age and gender. Yet, software programs have no difficulty to use nonsense determinants, and the investigator in charge of the analysis has to decide what is caused by what, because a computer can not do a thing like that, although it is essential to the analysis. It is our experience that a pocket calculator is very helpful for the purpose of studying the basic principles. Also, a number of statistical methods can be performed more easily on a pocket calculator, than using a software program. Advantages of the pocket calculator method include the following. 1. You better understand what you are doing. The statistical software program is kind of black box program. 2. The pocket calculator works faster, because far less steps have to be taken. 3. The pocket calculator works faster, because averages can be used. 4. With statistical software all individual data have to be included separately, a time-consuming activity in case of large data files. Also, some analytical methods, for example, power calculations and required sample size calculations are difficult on a statistical software program, and easy on
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Preface
a pocket calculator. The current book reviews the pocket calculator methods together with practical examples. This book was produced together with the similarly sized book “SPSS for Starters” from the same authors (edited by Springer, Dordrecht 2010). The two books complement one another. However, they can be studied separately as well. Lyon December 2010
Ton J. Cleophas Aeilko H. Zwinderman
Contents
1 Introduction..............................................................................................
1
2 Standard Deviations.................................................................................
3
3 t-Tests........................................................................................................ 1 Sample t-Test.......................................................................................... Paired t-Test............................................................................................... Unpaired t-Test...........................................................................................
5 5 6 7
4 Non-Parametric Tests.............................................................................. Wilcoxon Test............................................................................................ Mann-Whitney Test....................................................................................
9 9 10
5 Confidence Intervals................................................................................
15
6 Equivalence Tests.....................................................................................
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7 Power Equations......................................................................................
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8 Sample Size............................................................................................... Continuous Data, Power 50%.................................................................... Continuous Data, Power 80%.................................................................... Continuous Data, Power 80%, 2 Groups................................................... Binary Data, Power 80%............................................................................ Binary Data, Power 80%, 2 Groups...........................................................
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9 Noninferiority Testing.............................................................................. Step 1: Determination of the Margin of Noninferiority, the Required Sample, and the Expected p-Value and Power of the Study Result................................................................... Step 2: Testing the Significance of Difference Between the New and the Standard Treatment.........................................................
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Step 3: Testing the Significance of Difference Between the New Treatment and a Placebo.............................................................. Conclusion.................................................................................................
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10 Z-Test for Cross-Tabs..............................................................................
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11 Chi-Square Tests for Cross-Tabs............................................................ First Example Cross-Tab............................................................................ Chi-Square Table (c2-Table)...................................................................... Second Example Cross-Tab....................................................................... Example for Practicing 1........................................................................... Example for Practicing 2...........................................................................
31 31 31 33 33 34
12 Odds Ratios..............................................................................................
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13 Log Likelihood Ratio Tests......................................................................
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14 McNemar’s Tests ..................................................................................... Example McNemar’s Test.......................................................................... McNemar Odds Ratios, Example..............................................................
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15 Bonferroni t-Test...................................................................................... Bonferroni t-Test........................................................................................
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16 Variability Analysis.................................................................................. One Sample Variability Analysis............................................................... Two Sample Variability Test......................................................................
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17 Confounding.............................................................................................
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18 Interaction................................................................................................ Example of Interaction...............................................................................
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19 Duplicate Standard Deviation for Reliability Assessment of Continuous Data..................................................................................
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20 Kappas for Reliability Assessment of Binary Data...............................
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Final Remarks..................................................................................................
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Index..................................................................................................................
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Chapter 1
Introduction
This book contains all statistical tests that are relevant to starting clinical investigators. It begins with standard deviations and t-tests, the basic tests for the analysis of continuous data. Next, non-parametric tests are reviewed. They are, particularly, important to investigators whose affection towards medical statistics is little, because they are universally applicable, i.e., irrespective of the spread of the data. Then, confidence intervals and equivalence testing as methods based on confidence intervals are explained. In the next chapters power-equations that estimate the statistical power of data samples are reviewed. Methods for calculating the required sample size for a meaningful study, are the next subject. Non-inferiority testing including comparisons against historical data and sample size assessments are, subsequently, explained. The methods for assessing binary data include: z-tests, chi-square for cross-tabs, log likelihood ratio tests and odds ratio tests. Mc Nemar’s tests for the assessment of paired binary data is the subject of Chap. 14. Then, the Bonferroni test for adjustment of multiple testing is reviewed, as well as chi-square en F-tests for variability analysis of respectively one and two groups of patients. In the final chapters the assessment of possible confounding and possible interaction is assessed. Also reliability assessments for continuous and binary data are reviewed. Each test method is reported together with (1) a data example from practice, (2) all steps to be taken using a scientific pocket calculator, and (3) the main results and their interpretation. All of the methods described are fast, and can be correctly carried out on a scientific pocket calculator, such as the Casio fx-825, the Texas TI-30, the Sigma AK222, the Commodoor and many other makes. Although several of the described methods can also be carried out with the help of statistical software, the latter procedure will be considerably slower. In order to obtain a better overview of the different test methods each chapter will start on an uneven page. The pocket calculator book will be applied as a major help to the workshops “Designing and performing clinical research” organized by the teaching department of Albert Schweitzer STZ (collaborative top clinical)
T.J. Cleophas and A.H. Zwinderman, Statistical Analysis of Clinical Data on a Pocket Calculator: Statistics on a Pocket Calculator, DOI 10.1007/978-94-007-1211-9_1, © Springer Science+Business Media B.V. 2011
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1 Introduction
Hospital Dordrecht, and the statistics modules at the European College of Pharmaceutical Medicine, Claude Bernard University, Lyon, and Academic Medical Center, Amsterdam. The authors of this book are aware that it consists of a minimum of text and do hope that this will enhance the process of mastering the methods. Yet we recommend that for a better understanding of the test procedures the book be used together with the same authors’ textbook “Statistics Applied to Clinical Trials” 4th edition edited 2009, by Springer Dordrecht Netherlands. More complex data files like data files with multiple treatment modalities or multiple predictor variables can not be analyzed with a pocket calculator. We recommend that the in 2010 by the same editor published book “SPSS for Starters” (Springer, Dordrecht, 2010) from the same authors be used as a complementary help for the readers’ benefit.
The human brain excels in making hypotheses, but hypotheses have to be tested with hard data.
Chapter 2
Standard Deviations
Standard deviations (SDs) are often being used for summarizing the spread of the data from a sample. If the spread in the data is small, then the same will be true for the standard deviation. Underneath the calculation is illustrated with the help of a data example.
Mean
55 54 51 55 53 53 54 52+ =>
…/8 = 53.375
SD= 55 54 51 55 53 53 54 52 SD=
(55–53.375)2 (54–53.375)2 (51–53.375)2 (55–53.375)2 (53–53.375)2 (53–53.375)2 (54–53.375)2 (52–53.375)2+ ………… =>…./ n−1=> √….=> 1.407885953
Each scientific pocket calculator has a modus for data-analysis. It is helpful to calculate in a few minutes the mean and standard deviation of a sample.
T.J. Cleophas and A.H. Zwinderman, Statistical Analysis of Clinical Data on a Pocket Calculator: Statistics on a Pocket Calculator, DOI 10.1007/978-94-007-1211-9_2, © Springer Science+Business Media B.V. 2011
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2 Standard Deviations
Calculate standard deviation: mean = 53.375 SD = 1.407885953 The next steps are required: Casio fx-825 scientific On … mode … shift … AC … 55 … M+ … 54 … M+ … 51 … M+ … 55 … M+ … 53 … M+ … 53 … M+ … 54 … M+ … 52 … M+ … shift … [x] … shift … sxn–1 Texas TI-30 scientific On … 55 … S+ … 54 … S+ … 51 … S+ … 55 … S+ … 53 … S+ … 53 … S+ … 54 … S+ … 52 … S+ … 2nd … x … 2nd … sxn–1 Sigma AK 222 and Commodoor On … 2ndf … on … 55 … M+ … 54 … M+ … 51 … M+ … 55 … M+ … 53 … M+ … 53 … M+ … 54 … M+ … 52 … M+ … x=>M … MR Calculator: Electronic Calculator On … mode … 2 … 55 … M+ … 54 … M+ … 51 … M+ … 55 … M+ … 53 … M+ … 53 ... M+ … 54 … M+ … 52 … M+ … Shift … S-var … 1 … = … (mean) … Shift … S-var … 3 … (sd) Example: What is the mean value, what is de SD? 5 4 5 4 5 4 5 4
Chapter 3
t-Tests
1 Sample t-Test As an example, the mean decrease in blood pressure after treatment is calculated with the accompanying p-value. A p-value 95% chance that the decrease is the result of a real blood pressure lowering effect of the treatment. We call such a decrease statistically significant. Patient 1 2 3 4 5 6 7 8
mm Hg decrease 3 4 −2 3 1 −2 4 3
Is this decrease statistically significant? Mean decrease = 1.75 mmHg SD = 2.49 mmHg From the standard deviation the standard error (SE) can be calculated using the equation SE = SD / √ n
(n = sample size)
SE = 2.49 / √ 8 = 0.88 De t-value is the test-statistic of the t-test and is calculated as follows: t = 1.75 / 0.88 = 1.9886
T.J. Cleophas and A.H. Zwinderman, Statistical Analysis of Clinical Data on a Pocket Calculator: Statistics on a Pocket Calculator, DOI 10.1007/978-94-007-1211-9_3, © Springer Science+Business Media B.V. 2011
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3 t-Tests
Because the sample size is 8, the test has here 8−1 = 7 degrees of freedom. The t-table on the pages 7–8 shows that with 7 degrees of freedom the p-value should equal: 0.05
